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Meromorphic Continuation of Dynamical Zeta Functions via Transfer Operators J. Hilgert∗† F. Rilke‡§ Abstract 7 We describe a general method to prove meromorphic continuation of 0 dynamicalzetafunctionstotheentirecomplexplaneunderthecondition 0 2 thatthecorrespondingpartitionfunctionsaregivenviaadynamicaltrace formulafrom afamily oftransfer operators. Furtherwegive general con- n ditions for the partition functions associated with general spin chains to a be of this type and provide various families of examples for which these J conditions are satisfied. 3 Keywords: Dynamicalzetafunction,transferoperator,traceformulae, thermodynamic formalism, spin chain, Fock space, regularized determi- ] A nants,weighted composition operator. F . h Introduction t a m The dynamical zeta functions of interest in this paper are generating functions of the form [ 1 ∞ zn (1) ζ (z):=exp Z . v R n n 7 (cid:16)nX=1 (cid:17) 9 0 associated with sequences Zn n∈N in C. If the Zn arise as partition functions 1 of a dynamical system, the notion of dynamical zeta functions or Ruelle zeta 0 functions has been introdu(cid:0)ced(cid:1)by Ruelle in [21] and [22]. Naming and special 7 form of these functions are motivated by concrete examples from statistical 0 mechanics. Thepartitionfunctionencodesthe statisticalpropertiesofasystem / h in thermodynamic equilibrium. It depends on the temperature, the volume, t a and the microstates of a finite number of particles. We will consider partition m functions of the form : v n1 Xi (2) Zn = (−1)νtrace Gnν or Zn =det(1−Λn)trace Gn ν=0 X r a ∗Universita¨tPaderborn,Institutfu¨rMathematik,WarburgerStr. 100,D-33098Paderborn, Germany;[email protected] †correspondingauthor. ‡Universita¨tPaderborn,Institutfu¨rMathematik,WarburgerStr. 100,D-33098Paderborn, Germany;[email protected] §SupportedbytheDeutsche Forschungsmeinschaftthroughtheresearchprojects“Gitter- spinsysteme”and“GitterspinsystemeII”. 1 2 for large n. In the first case one has a (possibly) infinite family of compact operators G , called transfer operators, such that the corresponding Schatten ν norms satisfy nν=10kGnνokS1(Hν) <∞. In the secondcase one has as a transfer operator G together with an auxiliary operator Λ. We will refer to formulae P of the type (2) as dynamical trace formulae. Examples of such dynamical zeta functions derived fromtrace formulae of the type (2) have been treated repeat- edly in the literature, see e.g. [9], [11], [26], [27], [12], [15], [14], [4], [5], [18], [19]. Unlike other kinds of zeta functions such as Riemann’s, Selberg’s, or Artin’s zeta function, our dynamical zeta function is an exponential of a power series, hence itself a power series. Considering s ζ (e−s) one obtains a function R 7→ which is holomorphic in a right half plane, provided ζ has a non-zero radius R of convergence. Zeta functions typically occur as a kindof generatingfunctions forcollectionsofobjectslikeprimenumbersorprimegeodesicsanditisnatural to ask for their analytic properties. In particular, in order to prove asymptotic resultsforcountingfunctionsitisimportanttoknowwhetherthezetafunctions have meromorphic continuations to a larger set or even to the entire complex plane. Moreover, one wants to obtain information on the location of the poles. We will show that this kind of information can indeed be provided if the partition function can be written via dynamical trace formulae. More precisely, we show that such dynamical zeta functions are quotients of regularized Fredholm determinants, one factor for each transfer operator (see Theorem 2.1 and The- orem 2.3). Thus they have representationsas Euler products, and it is possible to give a spectral interpretation for its zeros and poles. Results of the kind sketched above become interesting only if one has a suffi- cient supply of examples of partition functions with the desired properties. We provideageneralprinciplehowtoconstructsuchexamplesfrom(classical)spin chains. To describe these, we consider a Hausdorff space F equipped with a finite measure ν, and let A : F F 0,1 be a ν ν-measurable function, whichwe callatransition matrix×. Th→en{ΩA :=} ξ F⊗N A(ξi,ξi+1)=1 i will { ∈ | ∀ } bereferredtoasaconfigurationspace. FurtherwefixaninteractionA b(ΩA). These data, together with the left shift τ :FN FN, (τξ) :=ξ ,∈arCe called k k+1 → a spin chain or, more technically, a one-sided one-dimensional matrix subshift. With such a matrix subshift we associate a dynamical partition function n n−1 (3) Z (A):= A exp A(τk(x ...x )) dνn(x ,...,x ), n xi,xi+1 1 n 1 n ZFnYi=1 (cid:16)kX=0 (cid:17) where x ...x := (x ,...,x ,x ,...,x ,...) and x := x . Consider the 1 n 1 n 1 n n+1 1 specialcaseofA=0,andletρ :FN Fn, ξ (ξ ,...,ξ )be theprojection. n 1 n → 7→ Then Zn(0)=νn ρn( ξ ΩA τnξ =ξ ) , { ∈ | } which measures the number of c(cid:0)losed τ-orbits in ΩA with(cid:1) period length n with respect to the a priori measure ν. In particular, if the system is a full shift, i.e., A 1, then Z (0) = ν(F)n. For general non-interacting matrix subshifts n ≡ it is possible to show (see Proposition 5.1) that there is an operator A such G that Z (0) = trace n. If the interaction A is non-zero we have to make more n A G assumptions in order to guarantee such trace representations of the partition function. SeeTheorem5.3forapreciseformulation. Itsproofdependsonatwo 3 special types of trace formulae. One (see Lemma 1.6) is elementary but subtle anddealswithiteratesofaveragesofHilbert-Schmidtoperators. Theother(see Theorem 4.6) deals with composition operators on Fock spaces and is based on a fixed point formula of Atiyah and Bott. In order to satisfy the hypotheses of Theorem 5.3 one has to verify certain estimates whichboildowntoaskingfor a ratherrapiddecayofthe interactions (see Theorem 6.2). Nevertheless, the theorem is strong enough to produce, among others, all the corresponding results scattered in the literature quoted above (see Example 6.5). The paper is organized as follows. In Section 1 we recall some key definitions and provide a number of technical results concerning traces and determinant in infinite dimensions used later in the paper. In Section 2 we show how to continuedynamicalzetafunctionsmeromorphicallyifadynamicaltraceformula holds. Section 3 contains a description of the composition operators that are instrumental in the construction of dynamical trace fromulae for spin chains in Section 5. The underlying trace formulae for these operators are proven in Section 4. In Section 6 we apply our results to Ising type spin chains and give examples for interactions for which our results can be applied. 1 Traces and Determinants GivenacompactoperatorK onaHilbertspace wewilldenoteby(λj(K))j∈N H the sequence of its eigenvalues counted with multiplicities. Let (sj(K))j∈N be the sequence of the singular numbers of K, i.e., the eigenvalues of the positive compact operator K = √K⋆K. For 1 p < the Schatten class ( ) is p | | ≤ ∞ S H defined as the space of all operators K such that kKkSp(H) :=k(sn(K))n∈Nkℓp(N) <∞. The Schatten classes ( ) End( ) for 1 p < are embedded subalge- p S H ⊂ H ≤ ∞ bras, i.e., they satisfy A A , AB A B , k kEnd(H) ≤k kSp(H) k kSp(H) ≤k kSp(H)k kSp(H) and have the approximation property, i.e., the finite rank operators are dense withrespectto (cf. [2,Thm. XI.11.1]). ForanyA ,...,A ( ) k·kSp(H) 1 ⌈p⌉ ∈Sp H one has (cf. [2, Thm. IV.11.2]) ⌈p⌉ (4) trace (A A ) A A A . | 1··· ⌈p⌉ |≤k 1··· ⌈p⌉kS1(H) ≤ k jkSp(H) j=1 Y By [2, Thm. XI.1.1] this estimate implies that for any n N the n - o ≥p o ∈ regularized determinant no−1 1 (5) det (1 F):=det(1 F) exp trace Fk no − − k (cid:16) kX=1 (cid:17) defined on finite rank operators admits a continuous extension to ( ), also p S H denoted by A det (1 A). The following assertions are true on the level 7→ no − of finite rank operators and can be extended to ( ) by continuity (see [2, p S H Thm. XI.2.1]). 4 Lemma 1.1. Let be a Hilbert space and 1 p n < . The function o H ≤ ≤ ∞ z det (1 zA)isentireforeveryfixedA ( )andhastherepresentation 7→ no − ∈Sp H ∞ c (A) (6) det (1 zA)=1+ n ( 1)nzn, no − n! − nX=no where the coefficients c (A) are defined by n b1 n−1 0 ... 0 0 b2 b1 n−2 ... 0 0 c (A):=det b3 ... ... ... ... ...  n  ... ... ... ... ... ...  bn−1 bn−2 bn−3 ... b1 1   bn bn−1 bn−2 ... b2 b1   and trace An, if n n , bn := 0, other≥wisoe. (cid:26) For z sufficiently small one has | | ∞ zn (7) det (1 zA) = exp trace An . no − − n (cid:16) nX=no (cid:17) Let (λ ) be the eigenvalues of A ( ), then one has the Euler product j j p ∈S H no−1λk (8) det (1 zA) = (1 zλ ) exp j zk = f (zλ ), no − − j k no j Yj (cid:16) (cid:0) kX=1 (cid:1)(cid:17) Yj no−1zk (⋆) ∞ zk where f (z)=(1 z) exp = exp . no − k − k (cid:16) kX=1 (cid:17) (cid:16) kX=no (cid:17) Theidentity(⋆)canbeobtainedasaconsequenceofthepowerseriesexpansion of log(1 z). The Euler product expansion (8) shows that the zeros of z − 7→ det (1 zA) are in bijection with the eigenvalues of A. no − Lemma 1.2. Let be a Hilbert space. Then for any n N there exists a o H ∈ constant Γ > 0 such that for all A ( ) the estimates below hold. One no ∈ Sno H can choose Γ =1. 1 (9) det (1+A) exp Γ Ano exp Γ A no . | no | ≤ nok kS1(H) ≤ nok kSno(H) Proof. The inequality det ((cid:0)1 + A) exp(cid:1)Γ A n(cid:0)o can be(cid:1)found in | no | ≤ nok kSno(H) [2, Thm. XI.2.2]. It based on the estimate |f(cid:0)no(z)| ≤ exp Γ(cid:1)no|z|no for some constant Γ >0. Using (8) one derives from this the slightly sharper estimate no (cid:0) (cid:1) det (1+A) exp Γ λ (A)no | no | ≤ no | j | j (cid:0) X (cid:1) = exp Γ λ (Ano) no | j | j (cid:0) X (cid:1) exp Γ Ano . ≤ nok kS1(H) (cid:0) (cid:1) 5 The following criterion for the convergence of infinite products of regularized determinants will turn out to be useful. Lemma 1.3. Let ( ν)ν∈N be a family of Hilbert spaces. Fix no N and pick Gν ∈Sno(Hν) satisfHying ∞ν=0kGnνokS1(Hν) <∞. Then ∈ P ∞ det (1 zG ). no − ν ν=0 Y converges absolutely and locally uniformly to an entire function of z. Proof. Note that the function f from Lemma 1.1 is of the form f (z) 1= no no − znohno(z) for some entire function hno. Further, c := supν∈N0kGνk is finite. Hence h (λ (zG )) sup h (w) =:c foralleigenvaluesλ (zG )of | no j ν |≤ |w|≤|z|c| no | z j ν zG . Now the hypothesis implies that ν ∞ ∞ f (λ (zG )) 1 λ (zG )no h (λ (zG )) | no j ν − | ≤ | j ν | | no j ν | ν=0 j ν=0 j XX XX ∞ c z no λ (Gno) ≤ z| | | j ν | ν=0 j XX ∞ c z no s (Gno) ≤ z| | | j ν | ν=0 j XX ∞ = c z no Gno z| | k ν kS1(Hν) ν=0 X is finite. Thus the infinite product ∞ f (λ (zG )) converges, and by ν=0 j no j ν Lemma 1.1 it is equal to ∞ det (1 zG ). This proves the claim. ν=0 noQ− Qν Proposition 1.4. Let Q and be two Hilbert spaces and n N. Pick 1 2 o H H ∈ A ( ) and B ( ). If we denote the eigenvalues of B by λ (B), ∈ Sno H1 ∈ Sno H2 j we have det 1 zA B = det 1 zλ (B)A . no − ⊗ no − j j (cid:0) (cid:1) Y (cid:0) (cid:1) Proof. For z < A −1 B −1 the Lidskii Trace Theorem ([2, Thm. | | k kSno(H1)k kSno(H2) 6 IV.6.1]) applied to the trace class operators Bn (n n ) yields o ≥ det (1 zA B) (=7) exp ∞ zn trace (A B)n no − ⊗ − n ⊗ (cid:16) nX=no (cid:17) ∞ zn = exp trace An trace Bn − n (cid:16) nX=no (cid:17) ∞ zn = exp trace An λ (B)n − n j (cid:16) nX=no Xj (cid:17) ∞ zn = exp λ (B)ntrace An − n j (cid:16) Xj nX=no (cid:17) ∞ zn = exp trace (λ (B)A)n − n j Yj (cid:16) nX=no (cid:17) (7) = det 1 zλ (B)A . no − j j Y (cid:0) (cid:1) Here we used that for z in the chosen range the inner double series converges absolutely and locally uniformly. By Lemma 1.1 the left hand side is an entire function in z. Thereforeanalytic continuationshows that the identity holds for all z C if we can show that also the righthand side is an entire function in z. ∈ But that follows from Lemma 1.3 applied to the family G :=λ (B)A. j j Let A : be a trace class operator on a Hilbert space and rA : r Hr→iHts r-fold exterior product. Then (cf. [23]) we haveH ∧ H→ H V V dimH (10) det(1 A)= ( 1)rtrace r A − − ∧ r=0 X and the estimate (11) rA 1 A r . k∧ kS1(∧rH) ≤ r!k kS1(H) ForthespecialcaseofafiniterankoperatorB withspectrumλ ,...,λ Propo- 1 d sition 1.4 implies that (12) det (1 zA νB)= det (1 zλαA), no − ⊗∧ no − α∈{0,1Y}d;|α|=ν where for α 0,1 d we set λα := d λαj. Approximating Schatten class ∈ { } j=1 j operators by finite rank operators one derives the following proposition: Q Proposition 1.5. For n N consider A ( ) and B ( ) with o ∈ ∈ Sno H1 ∈ Sno H2 eigenvalues λ :=λ (B). Then j j (i) det (1 zA νB)= lim det (1 zλαA). no − ⊗∧ d→∞ no − α∈{0,1Y}d;|α|=ν 7 (ii) ∞ det (1 zA νB)= lim det (1 zλαA). no − ⊗∧ d→∞ no − νY=0 α∈Y{0,1}d Proof. (i) For any d N let pr End( ) be the orthogonal projection ∈ d ∈ H2 onto the space spanned by the first d generalized eigenvectors of B and Bd := prd ◦B ◦prd. Then (Bd)d∈N converges to B in Sno(H2) by [2, Thm. IV 5.5]. Thus for any C ∈ Sno(H1) the sequence (C ⊗ Bd)d∈N converges to C B in ( ˆ ). The continuity of the regularized ⊗ Sno H1⊗H2 determinantX det (1 X) now shows lim det (1 zA B )= 7→ no − d→∞ no − ⊗ d det (1 zA B) for all A ( ), so that (12), applied to the B , no − ⊗ ∈ Sno H1 d implies the claim. (ii) We know that d det (1 zλαA) = det (1 zλαA) no − no − α∈Y{0,1}d νY=0α∈{0,1Y}d;|α|=ν d = det (1 zA νB ) no − ⊗∧ d ν=0 Y ∞ = det (1 zA νB ), no − ⊗∧ d ν=0 Y since νC = 0 for any C Mat(d,d;C) and ν > d. In view of (i) it only ∧ ∈ remains to show that ∞ ∞ lim det (1 zA νB )= lim det (1 zA νB ). d→∞ no − ⊗∧ d d→∞ no − ⊗∧ d ν=0 ν=0 Y Y ExpandingbothsidesintermsoftheeigenvaluesofBweseethatitsuffices to show ∞ ∞ det (1 zλαA) det (1 zλαA). no − d−→→∞ no − νY=0α∈{0,1Y}d;|α|=ν νY=0α∈{0,1Y}N;|α|=ν For this kind of rearrangement it is enough to verify the summability condition (16) from Lemma 1.3, i.e., the finiteness of ∞ ∞ zλαA = zA λα . k kSno(H) k kSno(H) | | Xν=0α∈{0,1X}N;|α|=ν Xν=0α∈{0,1X}N;|α|=ν To show this, we recall that the eigenvalues of the trace class operator 8 νB are the λα with α =ν and note that ∧ | | ∞ ∞ λα = λ ( νB) k | | | ∧ | νX=0α∈{0,1X}N;|α|=ν νX=0Xk ∞ s ( νB) k ≤ ∧ ν=0 k XX ∞ = ν B k∧ kS1(∧νH2) ν=0 X ∞ (11) 1 B ν < . ≤ ν!k kS1(H2) ∞ ν=0 X Lemma 1.6. Let (F,ν) be a measure space, g : F F C a measurable × → function, and (S ) a measurable family of operators on a separable Hilbert x x∈F space . The formula H (13) (T(f f ))(σ):= g(x,σ)f (x)S f dν(x) 1 2 1 x 2 ⊗ ZF defines a Hilbert-Schmidt operator T :L2(F,dν)ˆ L2(F,dν)ˆ iff ⊗H→ ⊗H (14) g(x,σ)2 dν(σ) S 2 dν(x)< . | | k xkS2(H) ∞ ZF ZF In this case T satisfies T 2 = g(x,σ)2 dν(σ) S 2 dν(x) k kS2(L2(F,dν)⊗ˆH) ZF ZF | | k xkS2(H) and n−1 trace Tn = g(x ,x ) g(x ,x )trace (S ... S )dνn(x ,...,x ) j j+1 n 1 xn◦ ◦ x1 1 n ZFn(cid:16)jY=1 (cid:17) for all n 2. Moreover, for these n we have ≥ n−1 Tn 2 = g(x ,x ) g(x ,σ) 2 k kS2(L2(F,dν)⊗ˆH) ZF ZFn(cid:12)(cid:16)jY=1 j j+1 (cid:17) n (cid:12) × S(cid:12) ... S 2 dνn(x ,(cid:12)...,x )dν(σ). ×k xn◦ ◦ x1kS2(H) 1 n Proof. Supposefirstthat(13)definesaHilbert-Schmidtoperator. Fixorthonor- malbases(ei)i∈N,(fj)j∈N forL2(F,dν)and ,respectively. Then,byParseval’s H 9 identity, one has ∞ T 2 = T(e f ) 2 k kS2(L2(F,dν)⊗ˆH) k i⊗ j k i,j=1 X ∞ 2 = T(e f ) e f i j k l ⊗ ⊗ i,jX,k,l=1(cid:12)(cid:12)(cid:10) (cid:11)(cid:12)(cid:12) ∞ (cid:12) (cid:12) 2 = g(x,σ)e (x) S f f dν(x)e (σ)dν(σ) i x j l k i,j∞X,k,l=1(cid:12)(cid:12)(cid:12)ZF ZF (cid:10) 2 (cid:11) (cid:12)(cid:12)(cid:12) = g(x,σ) S f f dν(x)dν(σ) x j l jX,l=1ZF ZF (cid:12)(cid:12) (cid:10) (cid:11)(cid:12)(cid:12) (cid:12) ∞ (cid:12) 2 = g(x,σ)2 S f f dν(x)dν(σ) x j l | | ZF ZF jX,l=1(cid:12)(cid:12)(cid:10) (cid:11)(cid:12)(cid:12) = g(x,σ)2 dν(σ(cid:12)) S 2 (cid:12)dν(x). | | k xkS2(H) ZF ZF Conversely,if (14)holds,wereversethis calculationandconcludethat notonly the integral (13) converges for almost all σ, but also that it defines a Hilbert- Schmidt operator on L2(F,dν)ˆ . ⊗H NowassumethatT isHilbert-Schmidt. Thenforn 2theoperatorTn istrace ≥ class and a simple induction argument shows that (Tn(e f))(σ)= ⊗ n−1 = g(x ,x ) g(x ,σ)e(x )S ... S fdνn(x ,...,x ). j j+1 n 1 xn ◦ ◦ x1 1 n ZFn(cid:16)jY=1 (cid:17) By the first part of the proof the S are Hilbert-Schmidt (for almost all x ), xj j hence the compositions S ... S are trace class. Now the trace of Tn can xn◦ ◦ x1 be calculated as follows ∞ trace Tn = Tn(e f ) e f i j i j ⊗ ⊗ i,j=1 X (cid:10) (cid:11) ∞ = g(x ,x ) g(x ,x )g(x ,σ) 1 2 n−1 n n ··· × i,j=1ZF ZFn X S ... S f f e (x )dνn(x ,...,x )e (σ)dν(σ) × xn ◦ ◦ x1 j j i 1 1 n i ∞ (cid:10) (cid:11) = g(x ,x ) g(x ,x )g(x ,σ) 1 2 n−1 n n ··· × i=1ZF ZFn X trace (S ... S )e (x )dνn(x ,...,x )e (σ)dν(σ). × xn ◦ ◦ x1 i 1 1 n i We claim that trace Tn can be rewritten as ∞ e e = trace with i=1 Gn i i Gn n−1 P (cid:10) (cid:11) ( f)(σ) := g(x ,x ) g(x ,σ) n j j+1 n G × ZFn(cid:16)jY=1 (cid:17) trace (S ... S )f(x )dνn(x ,...,x ). × xn ◦ ◦ x1 1 1 n 10 Note that (by Fourier expansion and induction) ∞ n−1 trace (S ... S ) = S h h S h h n◦ ◦ 1 j ij ij+1 n in i1 i1,.X..,in=1(cid:16)jY=1(cid:10) (cid:11)(cid:17) (cid:10) (cid:11) for any orthonormal basis (hi)i∈N for and Hilbert-Schmidt operators Si on H . Setting H ( f)(σ):= g(x,σ) S h h f(x)dν(x) i,j x i j G ZF for i, j N, we can rewrite as (cid:10) (cid:11) n ∈ G n−1 ( f)(σ) = g(x ,x ) g(x ,σ) n j j+1 n G × ZFn(cid:16)jY=1 (cid:17) ∞ n S h h f(x )dνn(x ,...,x ) × xi ij ij+1 1 1 n i1,.X..,in=1jY=1(cid:10) (cid:11) ∞ = ( ... f)(σ). Gin,i1 ◦Gin−1,in ◦ ◦Gi1,i2 i1,.X..,in=1 The identity ∞ ∞ (15) 2 = g(x,y)2 S h h 2dν(x)dν(y) kGi,jkS2(L2(F,dν)) | | | x i j | i,i=1 i,j=1ZF2 X X (cid:10) (cid:11) ∞ = g(x,y)2 S h h 2dν(x)dν(y) x i j | | | | ZF2 i,j=1 X (cid:10) (cid:11) = g(x,y)2 S 2 dν(x)dν(y) | | k xkS2(H) ZF2 implies thatthe areHilbert-Schmidt operatorsonL2(F,dν). Therefore,for i,j each (i ,...,i )G Nn the integral operator ... is trace 1 n ∈ Gin,i1 ◦Gin−1,in ◦ ◦Gi1,i2 classandby[7,Ex. X.1.18]itstracecanbeobtainedbyintegratingtheintegral kernel along the diagonal. If is trace class, we have n G ∞ trace = trace ( ... ) Gn Gin,i1 ◦Gin−1,in ◦ ◦Gi1,i2 i1,.X..,in=1 n−1 = g(x ,x ) g(x ,x )trace (S ... S )dνn(x ,...,x ) j j+1 n 1 xn ◦ ◦ x1 1 n ZFn(cid:16)jY=1 (cid:17) = trace Tn. Thus, to prove the claim it suffices to show that ∞ ... i1,...,in=1Gin,i1 ◦ ◦Gi1,i2 converges in L2(F,dν) . Using the technical Lemma 1.7 below, we obtain S1 P (cid:0) (cid:1)